Modeling and Forecasting S&P 500 Volatility: Long Memory, Structural Breaks and Nonlinearity

Abstract

The sum of squared intraday returns provides an unbiased and almost errorfree measure of ex-post volatility. In this paper we develop a nonlinear Autoregressive Fractionally Integrated Moving Average (ARFIMA) model for realized volatility, which accommodates level shifts, day-of-the-week effects, leverage effects and volatility level effects. Applying the model to realized volatilities of the S&P 500 stock index and three exchange rates produces forecasts that clearly improve upon the ones obtained from a linear ARFIMA model and from conventional time-series models based on daily returns, treating volatility as a latent variable.

Realized Volatility: A Review

“…More recently, Martens et al (2004) proposed a model that combines the long memory property with nonlinearity, which is especially important in modeling asymmetries and the leverage effect. They showed strong empirical evidence in favor of their proposal.…”

Realized Volatility Risk

“…N (0, 1), ψ t shifts the unconditional mean of realized volatility, d denotes the fractional differencing parameter, Φ(L) is a polynomial with roots outside the unit cirvle, L the lag operator, I is the indicator function, r j,t−1 is a notation for the cumulated returns Martens et al (2004), Thomakos and Wang (2003), among others. Nevertheless, the estimation of ARFIMA models in this context has encountered a few shortcomings.…”

“…1 Nevertheless, empirical work has found evidence of long range dependence even after accounting for possible regime changes and structural breaks in the volatility of asset returns (Lobato and Savin, 1998, Martens et al, 2004, Beltratti and Morana, 2006, Morana and Beltratti, 2004, Hyung and Franses, 2002, Scharth and Medeiros, 2009.…”

“…Martens et al (2004) and Scharth and Medeiros (2009) show that volatility sometimes tend to be lower on Mondays and Fridays, while substantially less volatility is observed around certain holidays.…”

“…Martens et al (2004) develop a nonlinear (ARFIMA) model to accommodate level shifts, day-of-the-week, leverage and volatility level effects. Andersen et al (2007) and Tauchen and Zhou (2005) argue that the inclusion of jump components significantly improves forecasting performance.…”